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Of the Lie Algebra Associated to the Lower Central transactions of the american mathematical society Volume 288. Number 1. March 1985 THE DETERMINATION OF THE LIE ALGEBRAASSOCIATED TO THE LOWER CENTRALSERIES OF A GROUP BY JOHN P. LABUTE1 This paper is dedicated to Professor Wilhelm Magnus Abstract. In this paper we determine the Lie algebra associated to the lower central series of a finitely presented group in the case where the defining relators satisfy certain independence conditions. Other central series, such as the lower p-central series, are treated as well. 1. Statement of results. The main purpose of this paper is to determine the Lie algebra associated to the lower central series of a group, thus extending the results of [6, 7] to groups defined by more than one relator. The methods apply to other central series such as the lower/?-central series. The lower central series of a group G is the sequence of subgroups Gn (n > 1) defined inductively by Gi = G> G«+i = [G'G«+i]> where [G, Gn + X]denotes the subgroup of G generated by the commutators [x, y] = x~1y~1xy with x e G, y e Gn. The associated graded abelian group gr(G) = 0n>1gr„(G), where gr„(G) = Gn/Gn + X, has the structure of a graded Lie algebra over the ring Z of integers, the bracket operation in gr(G) being induced by the commutator operation in G (cf. [2, 9, 11, 12]). The construction of the Lie algebra gr(G) uses only the fact that (G„) is a sequence of subgroups of G with the following properties: (i) Gx - G, (ii)G„+1cG„, (iii) [G„,GJcG„ +,. Such a family of subgroups of G is called a (central) filtration of G. Let F be the free group on the TVletters xx,...,xN, and let (F„) be the lower central series of F. If £, is the image of x¡ in gr,(F) = F/[F, F], then the Lie algebra Received by the editors May 10, 1983. Invited paper presented at the special session in Combinatorial Group Theory at the 803rd Meetingof the AMS in New York City, April 14-15,1983. 1980 Mathematics Subject Classification. Primary 20F14, 20F40; Secondary 20E18. Key words and phrases. Groups, defining relators, lower central series. Lie algebra. 1Supported by a grant from the National Science and Engineering Research Council of Canada. ©1985 American Mathematical Society 0002-9947/85 $1.00 + $.25 per page 51 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 52 J. P. LABUTE L = gr(F) associated to (Fn) is a free Lie algebra with basis £,,... ,£N (cf. loe. cit.). If x g F, x ¥= 1, there is a largest integer n = u(x) > 1 such that x g F„. This integer is called the weight of x (with respect to (F„)); the image of x in gr„(F) is called the initial form of x (with respect to (F„)). (If x = 1, its initial form is defined to be the zero element of L.) Let rx,...,rt G F and let p,,... ,p, be their initial forms with respect to the lower central series of F Let * = (p,,... ,pt) be the ideal of L generated by px,...,p, and let U be the enveloping algebra of L/i. Then */[*, *] is a iZ-module via the adjoint representation. Let # be the Lie algebra associated to the lower central series of G = F/R, where R = (rx,...,rf) is the normal subgroup of F generated by the elements rx,...,rr In general, g # L/i. unless the relators rx,...,r, satisfy certain independence conditions (cf. [7, 8] for the case t = 1). Theorem 1. If(i) L/i is a free Z-module and (ii) i/[i, i] is a free U-module on the images ofp,,... ,pt, then y = L/i. For any prime/? and any abelian group M we let M(p) = M/pM. If, in addition, A7is a subgroup of M, we let NM(p) be the image of N(p) in M(p). Conditions (i) and (ii) in Theorem 1 are equivalent to the following condition: (iii) For any prime p, iL(p)/[iL(p), *L(p)] is a free U(p)-module on the images of Pi.Pr In fact, condition (iii) implies that the rank of the nth homogeneous component of f(P) =f ® Z//?Z ¡s independent of p and hence that^ is a free Z-module (cf. [6, 7]). A formula for the rank of¿>„ can also be given (cf. loe. cit.). Example 1. r = N - 1, p, = fo, £2], p2 = [€2.É3L---.Pw-i = [Éw-i.íwl- Example 2. t = N - 1, Px = [£l5£2], p2 = [£l5 {3],...,pw_1 = [£lt £w]. Example 3. N = 3, t = 2, p, = [£3, [£x, |2]], p2 = [|2, [tlt |3]]. Example 4. 7V = 3, í = 2, Pl = [£lt £2], p2 = [[£,, £3], £2]. As a by-product of the proof, we obtain the following result: Theorem 2. Let T be the integral group ring of G, let I be the augmentation ideal of T, and let gr(T) = ®n>0I"/I" + l be the graded algebra associated to the I-adic filtration of V. Then, under the conditions (i) and (ii), we have gr(T) = U. Theorem 1 can be used to determine the structure of the lower central series quotients of certain link groups (cf. [3, 4, 5]). The proof of Theorem 1 requires the introduction of more general filtrations (see §2), and is proved in this more general context (see §3). The examples are treated in §4. In this section we also give an example to show that the theorem is not true under the hypotheses suggested in [12]. In §5 we obtain analogous results for the lower /?-central series. The above results are also true for pro-/?-groups with virtually the same proofs; one only has to replace Z by Zp (the ring of /?-adic integers), subgroups by closed subgroups, and the group ring by the completed group algebra over Z . For example, if G = F/R satisfies the conditions of Theorem 1, and G is the pro-/?-completion of G, then gr(G) = gr(G) ® Z . The techniques used in the proofs are contained in [6 and 7]; that they yield the above results does not seem to have been noticed. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE determination OF the lie algebra 53 2. The (x, T)-filtration of the free group F. Let A be the Magnus algebra of formal power series in the noncommutative indeterminates XX,...,XN with coefficients in Z. The homomorphism of F into the group of units of A defined by x¡ <-*1 + Xi is injective and can be extended to an injective homomorphism of the group ring A = Z[F] into A (see [2]). We identify A (and hence F) with its image in A. If tx,. .. ,tn are integers > 1, we define a valuation w of A by setting M I>, i X, ■■■Xi ) = Inf( t, + • • • + t, : a, , * O). For any integer n ^ 0 let An = {u g A: w(u) > n). Then A0 = A, An + X ç An and Am- AnQ Am+n which implies that A n is an ideal of A. Hence gr(A) = ®n>0grn(A), where gr„(^4) = An/An+X, has a natural structure of a graded ring. If £, is the image of X¡ in gr„(^4), where n = r¡, then gr(A) is the ring of noncommutative polynomials in the £, with coefficients in Z. If L is the Lie subring of gr(A) generated by the £„ then L is the free Lie algebra over Z with basis £,,... ,£N. If for n > 0 we set F„ = (1 + A„) n F, we obtain a filtration (F„) of F (cf. [2]). We call this filtration the (x, r)-filtration of F. If gr(F) is the Lie algebra associated to this filtration, the mapping F -» A defined by x <~*x — 1, induces an injective Lie algebra homomorphism of gr(F) into gr(yl), where the bracket operation in gr(A) is defined by [u, v] = uv — vu. We use this isomorphism to identify gr(F) with its image in gr(A). If gr(A) is the graded ring associated to the filtration (A„) of A, where A„ = An n A, then Lçgr(F)çgr(A)çgr(/l). It follows that gr(A) = gr(A). Let Tn (n > 1) be the set of elements of the form xf with e = ± 1, t, = 1, and define subsets Sn of F inductively as follows: Sx = Tx, and for n > 1, Sn = Tn U Tn', where Tn' is the set of elements of the form [x, y]e with e = ±1, x e S , y e S , p + q = n. Let Fn be the subgroup of F generated by the Sk with k > «. Then Fx = F, Fn+1 ç F„, and an easy calculation using the formulae (1) [x,yz] = [x,z][x,y][[x,y],z], (2) [xy,z] = [x,z][[x,z],y][x,y] shows that [F„, Fk] ç Fn + k. If t, = 1 for all i, then (F„) is the lower central series of F. If L is the Lie algebra associated to (F„), then the inclusions F„ Q F„ induce a canonical homomorphism of L into gr(F), which must necessarily be injective by the Poincaré-Birkhoff-Witt theorem since L is generated by lx,...,lN, where £,. is the image of x¡ in gr„(F) with n = t,. It follows that F„ = F„ for all n > 1, and hence thatL = L = gr(F).
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